Polar moment of inertia

Polar moment of inertia is a quantity used to predict an object's ability to resist torsion

Torsion (mechanics)

In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....

, in objects (or segments of objects) with an invariant circular cross section

Cross section (geometry)

In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

and no significant warping or out-of-plane deformation. It is used to calculate the angular displacement

Angular displacement

Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis....

of an object subjected to a torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

. It is analogous to the area moment of inertia, which characterizes an object's ability to resist bending

Bending

In engineering mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically...

and is required to calculate displacement

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

.

The larger the polar moment of inertia, the less the beam will twist, when subjected to a given torque.

Polar moment of inertia should not be confused with moment of inertia

Moment of inertia

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

, which characterizes an object's angular acceleration

Angular acceleration

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha .- Mathematical definition :...

due to a torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

. See moment (physics)

Moment (physics)

In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

Limitations

The polar moment of inertia cannot be used to analyze shafts with non-circular cross-sections. In such cases, the torsion constant

Torsion constant

The torsion constant is a geometrical property of a beam's cross-section which is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear-elastic bar. That is, the torsion constant describes a beam's torsional stiffness.- History...

can be substituted instead.

In objects with significant cross-sectional variation(along the axis of the applied torque), which cannot be analyzed in segments, a more complex approach may have to be used. See 3-D elasticity.

However the polar moment of inertia can be used to calculate the moment of inertia

Moment of inertia

of an object with arbitrary cross-section.

Definition

J_z = the polar moment of inertia about the axis z
dA = an elemental area
ρ = the radial distance to the element dA from the axis z

For a circular section with radius r:

Unit

The SI

Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

unit for polar moment of inertia, like the area moment of inertia, is metre to the fourth power (m⁴).

Conversion from Area Moment of Inertia

By the perpendicular axis theorem

Perpendicular axis theorem

In physics, the perpendicular axis theorem can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane...

, the following equation relates J_z to the area moments of inertia about the other two mutually perpendicular axes:

Application

NOTE: The following is only true for a circular or hollow circular section.

The polar moment of inertia appears in the formulae that describe torsional stress

Shear stress

A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

and angular displacement.

Torsional stress:

where

is the torque,

is the distance from the center and

is the polar moment of inertia.

In a circular shaft, the shear stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

is maximal at the surface of the shaft (as that is where the torque is maximal):

Most frequently the inverse problem is solved, in which one solves for the radius.

Sample calculation

Calculation of the steam turbine

Steam turbine

A steam turbine is a mechanical device that extracts thermal energy from pressurized steam, and converts it into rotary motion. Its modern manifestation was invented by Sir Charles Parsons in 1884....

shaft radius for a turboset:

Assumptions:

Power carried by the shaft is 1000 MW; this is typical for a large nuclear power
Nuclear power
Nuclear power is the use of sustained nuclear fission to generate heat and electricity. Nuclear power plants provide about 6% of the world's energy and 13–14% of the world's electricity, with the U.S., France, and Japan together accounting for about 50% of nuclear generated electricity...

plant.
Yield stress of the steel used to make the shaft (τ_yield) is: 250 x 10⁶ N/m².
Electricity has a frequency of 50 Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

; this is the typical frequency in Europe. In North America the frequency is 60 Hz.

The angular frequency

Angular frequency

In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

can be calculated with the following formula:

The torque carried by the shaft is related to the power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

by the following equation:

The angular frequency is therefore 314.16 rad

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

Second

The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

and the torque 3.1831 x 10⁶ N·m

Newton metre

A newton metre is a unit of torque in the SI system. The symbolic form is N m or N·m, and sometimes hyphenated newton-metre...

.

The maximal torque is:

After substitution of the polar moment of inertia the following expression is obtained:

The radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

is 0.200 m. If one adds a factor of safety

Factor of safety

Factor of safety , also known as safety factor , is a term describing the structural capacity of a system beyond the expected loads or actual loads. Essentially, how much stronger the system is than it usually needs to be for an intended load...

of 5 and re-calculates the radius with the maximal stress equal to the yield stress/5 the result is a radius of 0.343 m, or a diameter

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

of 69 cm, the approximate size of a turboset shaft in a nuclear power plant.

where:

d is the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
r is the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...